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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Examples</dfn>: 1. For</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation}
Y(s)=\frac{2 s^3+s^2+8s+6}{(s^2+1)(s^2+4)},\tag{8.3.1}
\end{equation}
</div>
<p class="continuation">using partial fractions, we can write <span class="process-math">\(Y(s)\)</span> in the form of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation}
Y(s)=\frac{as+b}{s^2+1}+\frac{cs+d}{s^2+4}=\frac{(as+b)(s^2+4)+(cs+d)(s^2+1)}{(s^2+1)(s^2+4)}.\tag{8.3.2}
\end{equation}
</div>
<p class="continuation">By expanding the numerator on the right hand side of (<a href="" class="xref" data-knowl="./knowl/Eq2_1.html" title="Equation 8.3.2">(8.3.2)</a>) and equating it to the numerator in (<a href="" class="xref" data-knowl="./knowl/Eq1_1.html" title="Equation 8.3.1">(8.3.1)</a>), we find that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation*}
2s^3+s^2+8s+6=(a+c)s^3+(b+d)s^2+(4a+c)s+(4b+d)
\end{equation*}
</div>
<p class="continuation">for all <span class="process-math">\(s\text{.}\)</span> Then, comparing coefficients of like powers of <span class="process-math">\(s\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation*}
a+c=2,\quad b+d=1,\quad 4a+c=8,\quad 4b+d=6.
\end{equation*}
</div>
<p class="continuation">Consequently, <span class="process-math">\(a=2, c=0, b=5/3, d=-2/3\text{,}\)</span> from which it follows that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation*}
Y(s)=\frac{2s}{s^2+1}+\frac{5/3}{s^2+1}-\frac{2/3}{s^2+4}.
\end{equation*}
</div>
<p class="continuation">Then from the table, the inverse Laplace transform of <span class="process-math">\(Y(s)\)</span> gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq2_1.html ./knowl/Eq1_1.html">
\begin{equation*}
y(t)=2 \cos t+\frac{5}{3} \sin t-\frac{1}{3} \sin 2t.
\end{equation*}
</div>
<span class="incontext"><a href="sec8_3.html#p-457" class="internal">in-context</a></span>
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